Hello Zaza and thanks for your good translation.
Here are my suggestions:
I would put a comma - Lady, smile
C'est la vie > That's life
parti means party, but not in our context
parti (=personne à marier) > match
ce beau parti très recherché > This good match much sought-after (much in demand)
madame à votre enfant > Lady to your (daughter) CHILD
l'amour ne chante bien qu'en liberté > Love sings well only freely
Kind regards
The remarks are gladly accepted.